2019/05/22 by Onti, Christos-Raent
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.09040
In this note, we investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let Mn be a complete conformally flat manifold and let f\colon Mn→ \Rm be an isometric immersion. We prove the following results: (1) If the index of relative nullity is at least two, then Mn is flat and f is a cylinder over a flat submanifold. (2) If the scalar curvature of Mn is non-negative and the index of relative nullity is positive, then f is a cylinder over a submanifold with constant non-negative sectional curvature. (3) If the scalar curvature of Mn is non-zero and the index of relative nullity is constant and equal to one, then f is a cylinder over a (n-1)-dimensional submanifold with non-zero constant sectional curvature.